However, the point that I am trying to make is that you cannot do that with bitcoin because you cannot lend out BTC that you do not have. If you start out with 100 BTC, you can never increase that amount by any kind of fractional reserve lending - there will only ever be 100 BTC.
The idea of "BTC credit" expressed as actual BTC makes no sense -- 1 BTC is 1 BTC it is the thing that is used for trade. If you want to create a new kind of currency that is backed by bitcoin and start a bank that uses that currency for fractional lending, then there is nothing to stop you, but you would have to operate in something other than actual BTC for credit - using actual BTC for that just won't work, as you'll quickly run out of any BTC that you had after only a few transactions/loans.
That's not exactly how fractional reserve banking works. Banks can lend out a multiple of the reserves they hold, but they can't lend out more than their total cash on hand. If $100 was all our hypothetical bank contained, then it could only lend out $90 of it, not $900.
The exact same thing could be done with Bitcoin. If you put it in a bank, they could lend this out to another party (and use future deposits to pay you back if you tried to withdraw your BTC).